Random Walks Crossing High Level Curved Boundaries

Let {S _n} be a random walk, generated by i.i.d. increments X _i which drifts weakly to in the sense that as n . Suppose k0, k1, and E|X ₁|^1\k = if k>1. Then we show that the probability that S. crosses the curve nan ^K before it crosses the curve n –an ^k tends to 1 as a . This intuitively plausible result is not true for k = 1, however, and for 1/2 <k<1, the converse results are not true in general, either. More general boundaries g(n) than g(n) = n ^k are also considered, and we also prove similar results for first passages out of regions like { (n, y): n1, |y| (a + n) ^k } as a .     

Certain problems of vector analysis

In the paper one gives an explicit method of constructing in the domain ^m m2 a vector field, having a prescribed divergencef in and taking prescribed values on the boundary . The differential properties of the field are faithfully determined by the smoothness off, and Simultaneously, one constructs the solutions of a series of other problems of vector analysis, which present an independent interest.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 65–85, 1984.     

Sur les Solutions D'un Opérateur Différentiel Singulier Semi-Linéaire

This paper deals with the following Dirichlet problem Lu = 1A ( Au – qu = – f ( , u ) on ] 0, [ , u, ( 0 ) = 0, u ( ) = 0, where ] 0, + ], q 0 is continuous on [ 0, [ × ] 0, + [ ] 0, + [ is continuous and A satisfies some appropriate conditions. The main result is the existence and the uniqueness of a strictly positive regular solution of the problem ( ). Moreover, we study the behaviour of this solution in a neighbourhood of . Our approach is based on the use of the Green's function of the homogeneous equation and Schauder's fixed point theorem.     

On strong solutions of the Stokes equations in exterior domains

We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -u + p=f inG, ·u=g inG, u= on in an exterior domainG ⁿ (n3) with boundary of class C². Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ·=g inG, = 0 on , the exterior domain problem is reduced to the entire space problem and an interior problem.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Theory of multiple j-elementary matrix-valued functions with a pole at the boundary of the unit circle

The theory of a matrix-valued function (), j-elementary in the unit circle and having a pole at the point ₀, ¦₀¦=1, on the boundary of the unit circle, is considered. The structure of () is determined, conditions for the splittingoff of () from an arbitrary matrix-valued function W(), j-expanding in ¦|< 1, are formed, a theorem on the parametrization of a j-elementary matrix-valued function () of full rank is proved, and a decomposition of () of full rank into the product of parametrized j-elementary factors of full rank with simple poles at the point ₀ is found.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 62–74, 1988.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

On (k,n)-blocking sets which can be obtained as a union of conics

A family of conics in PG(2,q) is called saturated if any line LPG(2,q) is incident with at least one conic of the family. Then, if <(q+1)/2, the support of is a (k,n)-blocking set. It is shown that in this way one can get blocking sets whose character n is small compared to q; it is also shown that cannot be taken independent of q, but must necessarily increase as q does.     

Waterman classes and spherical partial sums of double Fourier series

f(x,y) ₀BV(T²), ={1/n} _n=1 .

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Dedicated to Professor Károly Tandori, the outstanding mathematician and academician on his seventieth birthday

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This work was done under the financial support of the Russian Foundation for Fundamental Scientific Research, Grant 93-01-00240.     

The absolute convergence of weighted sums of dependent sequences of random variables

Summary The sum a _n X _n of a weighted series of a sequence {X _n } of identically distributed (not necessarily independent) random variables (r.v.s.) is a.s. absolutely convergent if for some in 0<1, ¦a _n ¦ < and E¦X _n ¦ < ; if a _n =z ⁿ for some ¦z¦<1 then it suffices that E(log¦X _n ¦)₊<. Examples show that these sufficient conditions are not necessary. For mutually independent {X _n } necessary conditions can be given: the a.s. absolute convergence of X _n z ⁿ (all ¦z¦<1) then implies E(log¦X _n ¦)₊ < , while if the X _n are non-negative stable r.v.s. of index , ¦a _n X _n ¦< if and only if ¦a _n ¦ < .     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Continuité de la densité de l'intégrale d'aire dans les espaces de Hardy

Nous estimons la dépendance en la fonction harmonique des fonctionnelles densité de l'intégrale d'aire. Nous obtenons notamment un contrôle en norme de sup _aR D u(,a)–D v(,a) ainsi qu'une inégalité aux bons- de type exponentielle.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Existence of compatible families of proper regular conditional probabilities

Summary Let (, , ) be a perfect probability space with countably generated, and let IB be a family of sub--fields of . Under a countability condition on the family IB, I show that there exists a family {}_IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.Research supported in part by NSF PHY-78-23952NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81).     

Hankel operators and problems of best approximation of unbounded functions

For each function f, f VMO, there exists a unique function f₀, analytic in the circle and such that f–f₀=_f{gVMO_A}. We define the operator of best approximation (nonlinear) A, Af=f₀, fVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X VMO, AX X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H², then AX X. For E one can take almost any space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985.     

Quasifractional Approximants in the Theory of Composite Materials

We study the effective heat conductivity of regular arrays of perfectly conducting spheres embedded in a matrix with unit conductivity. Quasifractional approximants allow us to derive an approximate analytical solution, valid for all values of the spheres volume fraction [0, _max] (_max is the maximum volume fraction of a spheres). As a starting point we use a perturbation approach for 0 and an asymptotic solution for _max. Three different spatial arrangements of the spheres, simple cubic, body centred and face centred cubic arrays, are considered. Results obtained give a good agreement with numerical data.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.